Mathematical Statements
Mathematical Statement
A meaningful composition of words which can be considered either true or false is called a mathematical statement or simply a statement.
A single letter shall be used to denote a statement. For example, the letter ‘p’ may be used to stand for the statement “ABC is an equilateral triangle.” Thus, p = ABC is an equilateral triangle.
Truth Value of a Statement
A statement is said to have truth value T or F according to whether the statement considered is true or false. For example, the statement ‘2 plus 2 is four’ has truth value T, whereas the statement ‘2 plus 2 is five’ has truth value F. The knowledge of truth value of statements enables us to replace one statement by another (equivalent) statement(s).
Production of New Statement
New statements from given statements can be produced by:
(i) Negation: $$ \sim $$
If p is a statement then its negation ‘$$ \sim $$p’ is statement ‘not p’. ‘$$ \sim $$p’ has truth value F or T according to the truth value of ‘p’ is T or F.
(ii) Implication: $$ \Rightarrow $$
If from a statement p another statement q follows, we say ‘p implies q’ and write ‘p$$ \Rightarrow $$ q’. Such a result is called an implication. The truth value of ‘p$$ \Rightarrow $$q’ is F only when p has truth value T and q has the truth value F.
The statements involving ‘if p holds then q’ are of the kind p$$ \Rightarrow $$q. For example, $$x = 2 \Rightarrow {x^2} = 4$$.
(iii) Conjunction: $$ \wedge $$
The sentence ‘p and q’ which may be denoted by ‘p$$ \wedge $$q’ is the conjunction of p and q. The truth value of p$$ \wedge $$q is T only when both p and q are true.
(iv) Disjunction: $$ \vee $$
The sentence ‘p and q (or both)’ which may be denoted by ‘p$$ \vee $$q’ is called the disjunction of the statements p and q. The truth value of p$$ \vee $$q is F only when both p and q are false.
Equivalence of Two Statements, p$$ \Leftrightarrow $$q
Two statements p and q are said to be equivalent if one implies the other, and in such a case we use the double implication symbol $$ \Leftrightarrow $$ and write p$$ \Leftrightarrow $$q.
The statements which involve the phrase ‘if and only if’ or ‘is equivalent to’ or ‘the necessary and sufficient conditions‘ are of the kind p$$ \Leftrightarrow $$q. For example, ABC is an equilateral triangle AB = BC = CA.
For brevity, the phrase ‘if and only if’ is shortened to “iff”. As described above, the symbols $$ \vee $$ stand for the words ‘and’ and ‘or’ respectively. The disjunction symbol $$ \vee $$ is used in the logical sense ‘and/or’. The symbols $$ \wedge $$, $$ \vee $$ are logical connectives and are frequently used.
The following is the table showing truth values of different compositions of statements. Such tables are called truth tables.
|
p
|
q
|
$$ \sim $$p
|
$$ \sim $$q
|
p$$ \Rightarrow $$q
|
p$$ \wedge $$q
|
p$$ \vee $$q
|
p$$ \Leftrightarrow $$q
|
|
T
|
T
|
F
|
F
|
T
|
T
|
T
|
T
|
|
T
|
F
|
F
|
T
|
F
|
F
|
T
|
F
|
|
F
|
T
|
T
|
F
|
T
|
F
|
T
|
F
|
|
F
|
F
|
T
|
T
|
T
|
F
|
F
|
T
|
By forming truth tables, the equivalence of various statements can easily be ascertained. For example, we shall easily see that the implication ‘p$$ \Rightarrow $$q’ is equivalent to ‘$$ \sim $$p$$ \Rightarrow $$$$ \sim $$q’. The implication ‘$$ \sim $$q$$ \Rightarrow $$$$ \sim $$p’ is called the contrapositive of p$$ \Rightarrow $$q.